The curve above will take some explaining. But first some other important clarifications:

Often, people don't distinguish between types of ice. So you'll hear them talk about 'ice is growing', when what they mean is the center of the Greenland ice cap, or Antarctic sea ice. My comment is specific to Arctic sea ice.

Another trap people fall in to is not paying attention to what sort of statement about ice is being made. There are two parts to this. I'm referring to sea ice extent, not area (well, at 0 extent you also have 0 area, but it's still something to keep in mind). Also, I'm referring to the monthly average for September. If some day showed zero ice cover before my 2035, give or take, that doesn't disprove the prediction. It takes a solid calendar month, September, of no ice to support or refute the prediction.

Then there's the fact that it's a probabilistic prediction. 2035 is the mid-point. By my estimation method, there's about a 50% chance (54%) that 2035 or some year before that will show zero ice extent for September. It's only 6% that we'd see zero ice in 2029 (or before) . And rises to 96% that we'll see zero ice (for the month) in 2042 or before. The 'or before' is important.

How I got to those predictions turns on the probability thing I mentioned in this year's sea ice estimation note, of it sometimes being easier to work with the probability of something not happening, than trying to figure out directly the chances of it happening.

The sea ice estimation note shows a best fit curve through the data and projects it in to the future. The fit has some fuzziness to it, meaning that it doesn't pass perfectly through the data points. We don't really expect it to either -- there is weather variability. The size of this mis-fit, or, conversely, weather variability, is about 0.45 million km^2.

If we pretend that the variability follows the Normal distribution, then we can use that and our predictions to estimate the probability that we'll see 0 ice in any given year. That doesn't turn out to be useful. For instance, suppose there was a 1% chance of seeing no ice this year, and 4% of seeing no ice next year. What's your chance of seeing no ice in

*either*year? I can compute it, but it's more tedious. It get rapidly more tedious with each year you want to consider. At 30-60 years ... ouch!

But we can consider quite easily the chance of there

*not*being ice this year

**and**there not being ice next year,

**and**the year after ... out however far you want to go. What makes this easy is the

**and**in the statement. If you're dealing with combining probabilities, and they combine by 'and', to find the chance of every single one happening, you just multiply the individual probabilities together. For the example from the previous paragraph, 1% chance of seeing no ice this year means 99% chance of

*seeing*ice this year. 3% chance of no ice, means 97% chance of ice. The probability, then of seeing ice in all (2) years is 0.99 * 0.97, for 0.96 (96%). We can then turn around and see that the chance of

*some*year not having ice is 100% - 96%, for 4%.

You might be noticing that 4% = 1% + 3%, and thinking you can just add the percentages. Consider tossing two coins. There's a 50% chance of getting heads on the first one, and 50% chance of getting heads on the second. (conversely, 50% chance of not getting heads). If you added, you would conclude that there's a 100% chance of not getting 2 heads. (50% chance of not getting a heads on the first coin

**and**50% chance of not getting heads on the second). The correct thing is to multiply (0.5*0.5), which says there's a 25% chance of getting no heads. Try tossing some pairs of coins and see which method is more correct.

In the top figure, I performed this kind of calculation for sea ice for every year out to 2070. As you see, the probability of not having any year with 0 ice for all of September is very high out to 2025. After that, we start accumulating some chances.

Now I can't say that I entirely believe this predictor. It is based only on my best fit logistic curve, rather than the ensemble. And this method knows nothing about sea ice thickness, only extent. But it makes a start on helping me (at least) think about what kinds of things are plausible. Given this, I'd need some good evidence behind a prediction of no ice for the month of September in, say, 2013. That seems extraordinarily unlikely. Conversely, 2060 looks extremely late (about a million to 1 against).

## 13 comments:

This is good stuff. You should quickly write a paper on it.

Is zero the most appropriate endpoint? The fast ice around Northern Greenland and the Canadian Archipelago will presumably be more resistant to melting out completely, just like the Antarctic shelves.

Thus, loss is likely to slow dramatically once all the floating ice has gone - do you have a prediction for that?

Bookmarked. Peter, if by fast ice you mean multiyear ice attached to the shores, there has been a very large reduction of the attachement points in recent years, and most of it flows like any other sea ice. The shores of the islands get warm during summers and free the multiyear ice to go with the flow. Of course ice shelves from glaciers on land will be there for a long time though some are disappearing in Antarctica.

Like Peter, I think you have to allow a small, but non-zero, amount of ice to persist on a longer time-frame than the majority of the ice.

Overall I think your time-frame is about right though. By 2030 the September ice, although still present, will be irrelevant to shipping.

Anon:

Any suggestions on where to submit it for publication?

All:

Good points.

Partly, you illustrate why it is I specified sea ice. The fast ice is not, properly speaking, sea ice. It also isn't included in the remote sensing that, for instance, the NSIDC does (too close to land).

Fast ice is its own interesting creature. I should take it up in its own post at some point.

Greg:

I'm not sure about the shipping question. For the Northeast passage, you might be right. The Northwest passage, though, should be less easy. The standard circulation pushes ice towards the Canadian Archipelago. Since we have recently opened, if only briefly, the Northwest passage, I'd expect the openings to become longer and more frequent. But the chance of ice getting blown into the passage would make it a concern for a long time.

So, in your last post, you listed three estimations: one based on statistical fit, one based on ensemble statistical fit, and one "Wu and Grumbine". Is this last one also a statistical fit, or is it something more complicated?

If this is only statistical fit work, then despite a long history of people publishing papers based on such things (in some cases famous - the Hubbert curve, for example - and in some cases infamous - various skeptic attempts to show climate sensitivity must be low by making an equation of T = CS*LN(CO2) + ENSO correction or whatever), I would advise against trying to publish in a serious journal. As your ensemble fit shows, there are a wide range of possible ways to fit a given curve shape to the data... and if you allow other curve shapes, why, as the saying goes, you could fit an elephant...

(but if Wu and Grumbine refers to something that is more complex than a straight logistic fit, then, go for it!)

-Marcus

I would like to add my voice to doing the same work with endpoint at 10% being 10% of 1979-2000 average (for example), and maybe also a run for 15%.

Reason are

1)-with 10 or 15 % of average, on a ecosystemic basis you can say the arctic is gone (or call it Aarctic as its something totally different, see recent book called "Eaarth"...)

2) Given the properties of the (inverted) logistic curve, with its long asymptote to 0, 10 or 15% might happen A LOT sooner than 0%, right?

Thanks for the exercice, very informative.

I really admire this, I mean it really looks interesting! Very nice research.

Seems like your definition of zero ice - that it lasts a solid calendar month - is much tougher than a 'zero ice at any one time during the year' that I've seen more often. The other definition could presumably happen much earlier.

Brian:

Certainly (as certain as I can be) we'll see a day of 0 ice cover before we see a solid month of 0 ice cover (and even longer before it is exactly a calendar month).

The thing is, the data I am working with is for monthly average extents. So the predictions have to be about that quantity. My approach could also be used for daily minimum extent, by anybody who looked up that data set. (Or constructed it -- the daily data do exist at http://nsidc.org/ but you might have to compute the extents yourself.) But I'm preparing for that summer break of my later post.

Daniel et alia:

I agree entirely that there are other interesting points to look for when we'll pass below. I'm a little leery of a figure like '10%', because that's suspiciously round and I don't think polar bears really know about 10%. This is another point, though, I'd like to get back to, along with the seasonal daily minima.

If you or anyone knows of research which sets and explains why a certain figure (whether it's 10% or 12.35% or 2.34 million square km, ...) is particularly meaningful to the climate system or some ecosystem, please do let me know what the figure is, and, of course, the source.

I dissed this in a throwaway comment on my blog, so maybe I owe you a longer version:

You only combine probabilities with "and" if they are independent (you know this). But the fact that you think they shift year-by-year rather suggests they aren't.

Also, your choice of normal distribution for the variability is trivially wrong in detail (because it gives a finite probability of less than zero ice). You can obviously patch that up, but the question of the distribution still matters. The chance of zero ice this year is zero, not a very small number. Ditto next year. Quite when it becomes non-zero I don't know.

> why a certain figure ... is ...

> meaningful to ... some ecosystem

Perhaps after break you could invite some of the ecologists who study the area to comment here, unless someone knows of places outside the journals this is discussed. Plenty studied, e.g. limit this to recent papers:

http://scholar.google.com/scholar?hl=en&q=polar+arctic+"ice-dependent"+ecosystem+primary+production

Pretty good post. I just stumbled upon your blog and wanted to say that I have really enjoyed reading your blog posts. Any way I'll be subscribing to your feed and I hope you post again soon.

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